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InteractionsbetweenIntrinsicandStimulus­Evoked

ActivityinRecurrentNeuralNetworks

L.F.AbbottandKanakaRajan

DepartmentofNeuroscienceDepartmentofPhysiologyandCellularBiophysics

ColumbiaUniversityCollegeofPhysiciansandSurgeonsNewYork,NY10032‐2695USA

and

HaimSompolinsky

RacahInstituteofPhysicsInterdisciplinaryCenterforNeuralComputation

HebrewUniversityJerusalem,Israel

Introduction

Trial‐to‐trialvariabilityisanessentialfeatureofneuralresponses,butitssourceisasubjectofactivedebate.Responsevariability(MastandVictor,1991;Arielietal.,1995&1996;Andersonetal.,2000&2001;Kenetetal.,2003;Petersenetal.,2003a&b;Fiser,ChiuandWeliky,2004;MacLeanetal.,2005;Yusteetal.,2005;Vincentetal.,2007)isoftentreatedasrandomnoise,generatedeitherbyotherbrainareas,orbystochasticprocesseswithinthecircuitrybeingstudied.Wecallsuchsourcesofvariability“external”tostresstheindependenceofthisformofnoisefromactivitydrivenbythestimulus.Variabilitycanalsobegeneratedinternallybythesamenetworkdynamicsthatgeneratesresponsestoastimulus.Howcanwedistinguishbetweenexternalandinternalsourcesofresponsevariability?Hereweshowthatinternalsourcesofvariabilityinteractnonlinearlywithstimulus‐inducedactivity,andthisinteractionyieldsasuppressionofnoiseintheevokedstate.Thisprovidesatheoreticalbasisandpotentialmechanismfortheexperimentalobservationthat,inmanybrainareas,stimulicausesignificantsuppressionofneuronalvariability(WernerandMountcastle,1963;Fortier,SmithandKalaska,1993;Andersonetal.,2000;FriedrichandLaurent,2004;Churchlandetal.,2006;Finn,PriebeandFerster,2007;Mitchell,SundbergandReynolds,2007;Churchlandetal.,2009).Thecombinedtheoreticalandexperimentalresultssuggestthatinternallygeneratedactivityisasignificantcontributortoresponsevariabilityinneuralcircuits.

Weareinterestedinuncoveringtherelationshipbetweenintrinsicandstimulus‐evokedactivityinmodelnetworksandstudyingtheselectivityofthesenetworkstofeaturesofthestimulidrivingthem.Therelationshipbetweenintrinsicand

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extrinsicallyevokedactivityhasbeenstudiedexperimentallybycomparingactivitypatternsacrosscorticalmaps(Arielietal.,1995&1996).Wedeveloptechniquesforperformingsuchcomparisonsincaseswherethereisnoapparentsensorymap.Inadditiontorevealinghowthetemporalandspatialstructureofspontaneousactivityaffectsevokedresponses,thesemethodscanbeusedtoinferinputselectivity.Historically,selectivitywasfirstmeasuredbystudyingstimulus‐drivenresponses(HubelandWiesel,1962),andonlylaterweresimilarselectivitypatternsobservedinspontaneousactivityacrossthecorticalsurface(Arielietal.,1995&1996).Wearguethatitispossibletoworkinthereverseorder.Havinglittleinitialknowledgeofsensorymapsinournetworks,weshowhowtheirspontaneousactivitycaninformusabouttheselectivityofevokedresponsestoinputfeatures.Throughoutthisstudy,werestrictourselvestoquantitiesthatcanbemeasuredexperimentally,suchasresponsecorrelations,soouranalysismethodscanbeappliedequallytotheoreticalmodelsandexperimentaldata.

Webeginbydescribingthenetworkmodelandillustratingthetypesofactivityitproduces,usingcomputersimulations.Inparticular,weillustrateanddiscussatransitionbetweentwotypesofresponses;oneinwhichintrinsicandstimulus‐evokedactivitycoexist,andtheotherinwhichintrinsicactivityiscompletelysuppressed.Next,weexplorehowthespatialpatternsofspontaneousandevokedresponsesarerelated.Byspatialpattern,wemeanthewaythatactivityisdistributedacrossthedifferentneuronsofthenetwork.Spontaneousactivityisausefulindicatorofrecurrenteffects,becauseitiscompletelydeterminedbynetworkfeedback.Therefore,westudytheimpactofnetworkconnectivityonthespatialpatternofinput‐drivenresponsesbycomparingthespatialstructureofevokedandspontaneousactivity.Finally,weshowhowthestimulusselectivityofthenetworkcanbeinferredfromananalysisofitsspontaneousactivity.

TheModel

Neuronsinthemodelweconsideraredescribedbyfiring‐rates,theydonotfireindividualactionpotentials.Suchfiring‐ratenetworksareattractivebecausetheyareeasiertosimulatethanspikingnetworkmodelsandareamenabletomoredetailedmathematicalanalyses.Ingeneral,aslongasthereisnolarge‐scalesynchronizationofactionpotentials,firing‐ratemodelsdescribenetworkactivityadequately(Shriki,HanselandSompolinsky,2003;WongandWang,2006).WeconsideranetworkofNinterconnectedneurons,withneuronicharacterizedbyanactivationvariablexisatisfying

€

τdxidt

= −xi + g Jijrj + Iij=1

N

∑ .

Thetimeconstantτissetto10ms.Forallofthefigures,except5e,N=1000.TherecurrentsynapticweightmatrixJhaselementJijdescribingtheconnectionfrompresynapticneuronjtopostsynapticneuroni.Excitatoryconnectionscorrespond

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topositivematrixelements,inhibitoryconnectionstonegativeelements.Theinputterm,Iiforneuroni,takesvariousformsthatwillbedescribedasweusethem.

Thefiringrateofneuroniisgivenby

€

ri = R0 + φ(x i) with

€

φ(x) = R0 tanh(x /R0)for

€

x ≤ 0 and

€

φ(x) = (Rmax − R0)tanh(x /(Rmax − R0)) for

€

x > 0 .Here,R0isthebackgroundfiringrate(thefiringratewhenx=0),andRmaxisthemaximumfiringrate.Thisfunctionallowsustospecifyindependentlythemaximumfiringrate,Rmax,andthebackgroundrate,R0,andsetthemtoreasonablevalues,whileretainingthegeneralformofthecommonlyusedtanhfunction.ThisfiringratefunctionisplottedinFigure1forR0=0.1Rmax,thevalueweuse.Tofacilitatecomparisonwithexperimentaldatainavarietyofsystems,wereportallresponsesrelativetoRmax.Similarly,wereportallinputcurrentsrelativetothecurrentI1/2requiredtodriveanisolatedneurontohalfofitsmaximalfiringrate(seeFigure1).

Figure1.ThefiringratefunctionusedinthenetworkmodelversustheinputIforR0=0.1RmaxnormalizedbythemaximumfiringrateRmax.TheparameterI1/2isdefinedbythedashedlines.

Althoughaconsiderableamountisknownaboutthestatisticalpropertiesof

recurrentconnectionsincorticalcircuitry(Holmgrenetal.,2003;Songetal.,2005),wedonothaveanythinglikethespecificneuron‐to‐neuronwiringdiagramwewouldneedtobuildatrulyfaithfulmodelofacorticalcolumnorhypercolumn.Instead,weconstructtheconnectionmatrixJofthemodelnetworkonthebasisofastatisticaldescriptionoftheunderlyingcircuitry.WedothisbychoosingelementsofthesynapticweightmatrixindependentlyandrandomlyfromaGaussiandistributionwithzeromeanandvariance1/N.Wecoulddividethenetworkintoseparateexcitatoryandinhibitorysubpopulations,butthisdoesnotqualitativelychangethenetworkpropertiesthatwediscuss(vanVreeswijkandSompolinsky,1996&1998;RajanandAbbott,2006).

Theparametergcontrolsthestrengthofthesynapticconnectionsinthemodel,butbecausethesestrengthsarechosenfromadistributionwithzeromeanandnonzerovariance,gactuallycontrolsthesizeofthestandarddeviationofthesynapticstrengths(seeDiscussion).Withoutanyinput(Ii=0foralli)andforlargenetworks(largeN),twospontaneouspatternsofactivityareseen.Ifg<1,thenetworkisinatrivialstateinwhichxi=0andri=R0forallneurons(alli).Thecaseg>1ismoreinterestinginthatthespontaneousactivityofthenetworkischaotic,meaningthatitisirregular,non‐repeatingandhighlysensitivetoinitialconditions

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(Sompolinsky,Crisanti,andSommers,1988;vanVreeswijkandSompolinsky,1996&1998).Wetypicallyuseavalueofg=1.5,meaningthatournetworksareinthischaoticstatepriortoactivatinganyinputs.

Figure2.Firingratesandresponsevariabilitynormalizedbythemaximumfiringrate,Rmax,forastimulusinputsteppingfromzerotoaconstant,non‐zerovalueatt=1000ms.Leftcolumnshowsthefiringrateofatypicalneuroninanetworkwith1000neurons.Rightcolumnshowstheaveragefiringrate(redtraces)andthesquarerootoftheaveragefiring‐ratevariance(blacktraces)acrossthenetworkneurons.a‐b)Anetworkwithchaoticspontaneousactivityreceivingnonoiseinput.Theresponsevariability(b,blacktrace)dropstozerowhenthestimulusinputispresent.c‐d)Anetworkwithoutspontaneousactivitybutreceivingnoiseinput.Theresponsevariability(d,blacktrace)risesslightlywhenthestepinputisturnedon.Thestochasticinputinthisexamplewasindependentwhite‐noisetoeachneuron,low‐passfilteredwitha500mstimeconstant.ResponsestoStepInputTobeginourexaminationoftheeffectsofinputonchaoticspontaneousnetworkactivity,weconsidertheeffectofastepofinput(from0toapositivevalue),applieduniformlytoeveryneuron(Ii=Iforalli).Beforetheinputisturnedon(Figure2a&b,t<1000ms),atypicalneuronofthenetworkshowsthehighlyirregularactivitycharacteristicofthechaoticspontaneousstate.However,whenasufficientlystrongstimulusisapplied,theinternallygeneratedfluctuationsarecompletelysuppressed(Figure2a&b,t>1000ms).Wecontrastthisbehaviortothatofexternalnoise,byturningofftherecurrentdynamicsandgeneratingfluctuationswithexternal

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stochasticinputs(Figure1c&d,t<1000ms).Inthiscase,thereisnoreductionintheamplitudeoftheneuronalfluctuationswhenastimulusisapplied,infactthereisasmallincrease.Notethattheincreaseinthemeanactivityissimilarinbothcases.Theseresultsrevealacriticaldistinctionbetweeninternallyandexternallygeneratedfluctuations–theformercanbesuppressedbyastimulusandthereforedonotnecessarilyinterferewithsensoryprocessing.

Torevealthenatureofinternallygeneratedvariability,wehaveconsideredanidealizedscenarioinFigure2inwhichtherewasnoexternalsourceofnoise.Inreality,weexpectbothexternalandinternalsourcesofnoisetocoexistinlocalcorticalcircuits.Aslongastheinternalnoiseprovidesasubstantialcomponentoftheoverallvariability,ourqualitativeresultsremainvalid.Wecansimulatethissituationbyaddingexternalnoise(asinFigure2c&d)toamodelthatexhibitschaoticspontaneousactivity(asinFigure2a&b).Theresultshowsasharpdropinvarianceatstimulusonset,butwithonlypartial,ratherthancomplete,suppressionofresponsevariability(Figure3).Thisresultisingoodagreementwithexperimentaldata(Churchlandetal.,2009).

Figure3.Firingratesandresponsevariabilitynormalizedbythemaximumfiringrate,Rmax,forthesamenetwork,stimulusandnoiseasinFigure2,butforanetworkwithbothspontaneousactivityandinjectednoise.a)Theresponseofatypicalneuron.b)Theaveragefiringrate(redtrace)increasesandtheresponsevariability(blacktrace)decreaseswhenthestimulusinputispresent,butitdoesnotgotozeroasinFigure2b.

ResponsetoPeriodicInputForFigures2&3,thestimulusconsistedofastepinputappliedidenticallytoallneurons.Toinvestigatetheeffectofmoreinterestingandrealisticstimulionthechaoticactivityofarecurrentnetwork,weconsiderinputswithnon‐homogeneousspatio‐temporalstructure.Specifically,weintroduceinputsthatoscillateinasinusoidalmannerwithamplitudeIandfrequencyfandexaminehowthesuppressionoffluctuationsdependsontheiramplitudeandfrequency.Inmanycases,neuronsinalocalpopulationhavediversestimulusselectivities,soa

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particularstimulusmayinducelittlechangeinthetotalactivityacrossthenetwork.Tomimicthissituation,wegivetheseoscillatinginputsadifferentphaseforeachneuron(intermsofavisualstimulus,thisisequivalenttopresentingastationary,counterphasegratingtoapopulationofsimplecellswithdifferentspatial‐phaseselectivities).Specifically,Ii=Icos(2πft+θi),whereθiischosenrandomlyfromauniformdistributionbetween0and2π.Therandomlyassignedphasesensurethatthespatialpatternofinputinourmodelnetworkisnotcorrelatedwiththepatternofrecurrentconnectivity.

Figure4.Achaoticnetworkof1000neuronsreceivingsinusoidal5Hzinput.a)Firingratesoftypicalnetworkneurons(normalizedbyRmax).b)Thelogarithmofthepowerspectrumoftheactivityacrossthenetwork.i)Withnoinput(I=0),networkactivityischaotic.ii)Inthepresenceofaweakinput(I/I1/2=0.1),anoscillatoryresponseissuperposedonchaoticfluctuations.iii)Forastrongerinput(I/I1/2=0.5),thenetworkresponseisperiodic.

Intheabsenceofastimulusinput,thefiringratesofindividualneuronsfluctuateirregularly,asseeninFigure2(Figure4a‐i),andthepowerspectrumacrossnetworkneuronsiscontinuousanddecaysexponentiallyasafunctionoffrequency(Figure4b‐i),acharacteristicfeaturesofthechaoticstateofthisnetwork(Sompolinsky,Crisanti,andSommers,1988).Whenthenetworkisdrivenbyaweakoscillatoryinput,thesingle‐neuronresponseisasuperpositionofaperiodicpatterninducedbytheinputandachaoticbackground(Figure4a‐ii).Thepowerspectrumshowsacontinuouscomponentduetotheresidualchaosandpeaksatthefrequencyoftheinputanditsharmonics,reflectingtheperiodicbutnon‐sinusoidalcomponentoftheresponse(Figure4b‐ii).Foraninputwithalargeramplitude,thefiringratesofnetworkneuronsareperiodic(Figure4a‐iii),andthepowerspectrumshowsonlypeaksattheinputfrequencyanditsharmonics,withnocontinuous

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spectrum(Figure4b‐iii).ThisindicatesacompletesuppressionoftheinternallygeneratedfluctuationsasinFigure2a&b.

Wehaveusedamean‐fieldapproachsimilartothatdevelopedbySompolinsky,Crisanti,andSommers(1988)toanalyzepropertiesofthetransitionbetweenchaoticaperiodicresponsestoaperiodicstimulus(Rajan,AbbottandSompolinsky,2009).Thisextendspreviousworkontheeffectofinputonchaoticnetworkactivity(Molgedey,SchuchhardtandSchuster,1992;BertchingerandNatschläger,2004)tocontinuoustimemodelsandperiodicinputs.Wefindthatthereisacriticalinputintensity(acriticalvalueofI)thatdependsonfandg,belowwhichnetworkactivityischaoticthoughdrivenbytheinput(asinFigures4a‐ii&4b‐ii)andabovewhichitisperiodic(asinFigures4a‐iii&4b‐iii).Asurprisingfeatureofthiscriticalamplitudeisthatitisanon‐monotonicfunctionofthefrequencyfoftheinput.Asaresult,thereisa“best”frequencyatwhichitiseasiesttoentrainthenetworkandsuppresschaos.Fortheparametersweuse,the“best”frequencyisaround5Hz,afrequencyweremanysensorysystemstendtooperate,andtherearesomeinitialexperimentalindicationsthatthisisindeedtheoptimalfrequencyforsuppressingbackgroundactivitybyvisualstimulation(WhiteandFiser,2008).Itisinterestingthatapreferredinputfrequencyforentrainmentariseseventhoughthepowerspectrumofthespontaneousactivitydoesnotshowanyresonantfeatures(Figure4b‐i).

PrincipalComponentAnalysisofSpontaneousandEvokedActivityTheresultsoftheprevioustwosectionsrevealedaregimeinwhichaninputgeneratesanon‐chaoticnetworkresponse,eventhoughthenetworkischaoticintheabsenceofinput.Althoughthechaoticintrinsicactivityhasbeencompletelysuppressedinthisnetworkstate,itsimprintcanstillbedetectedinthespatialpatternofthenon‐chaoticactivity.

ThenetworkstateatanyinstantcanbedescribedbyapointinanN‐dimensionalspacewithcoordinatesequaltothefiringratesoftheNneurons.Overtime,activitytraversesatrajectoryinthisN‐dimensionalspace.Principalcomponentanalysiscanbeusedtodelineatethesubspaceinwhichthistrajectorypredominantlylies.Theanalysisisdonebydiagonalizingtheequal‐timecross‐correlationmatrixofnetworkfiringrates,<ri(t)rj(t)>,wheretheanglebracketsdenoteanaverageovertimeTheeigenvaluesofthismatrixexpressedasafractionoftheirsum(denotedby

€

˜ λ a ),indicatethedistributionofvariancesacrossdifferentorthogonaldirectionsintheactivitytrajectory.Inthespontaneousstate,thereareanumberofsignificantcontributorstothetotalvariance,asindicatedinFigure5a.Forthisvalueofg,theleading10%ofthecomponentsaccountfor90%ofthetotalvariance.Thevarianceassociatedwithhighercomponentsfallsoffexponentially.Itisinterestingtonotethattheprojectionsofthenetworkactivityontotheprincipalcomponentdirectionsfluctuatemorerapidlyforhighercomponents(Figure5c),revealingtheinteractionbetweenthespatialandtemporalstructureofthechaoticfluctuations.

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Figure5:Principalcomponentanalysisofthechaoticspontaneousstateandnon‐chaoticdrivenstate.a)Percentvarianceaccountedforbydifferentprincipalcomponentsforchaoticspontaneousactivity.b)Sameasa,butfornon‐chaoticdrivenactivity.c)Projectionsofthechaoticspontaneousactivityontoprincipalcomponentvectors1,10and50(indecreasingorderofvariance).d)Projectionsofperiodicdrivenactivityontoprincipalcomponents1,3,and5.Projectionsontocomponents2,4,and6aresimilarexceptforbeingphaseshiftedbyπ/2.e)Theeffectivedimension,Neff,ofthetrajectoryofchaoticspontaneousactivity(definedinthetext)asafunctionofgfornetworkswith1000(solidcircles)or2000(opencircles)neurons.Parameters:g=1.5fora‐d,andf=5andI/I1/2=0.7forbandd.

Thenon‐chaoticdrivenstateisapproximatelytwodimensional(Figure5b),with

thetwodimensionsdescribingacircularoscillatoryorbit.Projectionsofthisorbitcorrespondtotheoscillationsπ/2apartinphase.Theresidualvarianceinthehigherdimensionsreflectshigherharmonicsarisingfromnetworknonlinearity,asillustratedbytheprojectionsinFigure5d.

Toquantifythedimensionofthesubspacecontainingthechaotictrajectoryinmoredetail,weintroducethequantity

€

Neff = ˜ λ a2

a=1

N

∑

−1

.

Thisprovidesameasureoftheeffectivenumberofprincipalcomponentsdescribingatrajectory.Forexample,ifnprincipalcomponentssharethetotalvarianceequally,andtheremainingN­nprincipalcomponentshavezerovariance,Neff=n.Forthechaoticspontaneousstateinthenetworkswestudy,Neffincreaseswithg(Figure5e),duetothehigheramplitudeandfrequencycontentofthechaoticactivityforlargeg.NotethatNeffscalesapproximatelywithN,whichmeansthatlargenetworkshaveproportionallyhigher‐dimensionalchaoticactivity(comparethetwotracesinFigures5e).Thefactthatthenumberofactivatedmodesisonly2%ofthesystemdimensionality,evenforgashighas2.5,isanothermanifestationofthedeterministicnatureofthefluctuations.Forcomparison,wecalculatedNeffforasimilarnetworkdrivenbyexternalwhitenoise,withgsetbelowthechaotictransitionatg=1.Inthiscase,Neffonlyassumesuchlowvalueswhengiswithinafewpercentofthecriticalvalue1.TheresultsinFigure5illustrateanotherfeature

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ofthesuppressionofspontaneousactivitybyinput,whichisthatthePCAdimensionNeffisreduceddramaticallybythepresenceoftheinput.

NetworkEffectsontheSpatialPatternofEvokedActivityInthenon‐chaoticregime,thetemporalstructureofnetworkresponsesislargelydeterminedbytheinput;theybothoscillateatthesamefrequency,althoughthenetworkactivityincludesharmonicsnotpresentintheinput.Theinputdoesnot,however,exertnearlyasstrongcontrolonthespatialstructureofthenetworkresponse.Thephasesofthefiring‐rateoscillationsofnetworkneuronsareonlypartiallycorrelatedwiththephasesoftheinputsthatdrivethem,andtheyarestronglyinfluencedbytherecurrentfeedback.

Wehaveseenthattheorbitdescribingtheactivityinthenon‐chaoticdrivenstateconsistsprimarilyofacircleinatwo‐dimensionalsubspaceofthefullN‐dimensionsdescribingneuronalactivities.Wenowaskhowthiscirclealignsrelativetosubspacesdefinedbydifferentnumbersofprincipalcomponentsthatcharacterizethespontaneousactivity.Thisrelationshipisdifficulttovisualizebecauseboththechaoticsubspaceandthefullspaceofnetworkactivitiesarehighdimensional.Toovercomethisdifficulty,wemakeuseofthenotionof“principalangles”betweensubspaces(IpsenandMeyer,1995).

Thefirstprincipalangleistheanglebetweentwounitvectors(calledprincipalvectors),oneineachsubspace,thathavethemaximumoverlap(dotproduct).Higherprincipalanglesaredefinedrecursivelyastheanglesbetweenpairsofunitvectorswiththehighestoverlapthatareorthogonaltothepreviouslydefinedprincipalvectors.Specifically,fortwosubspacesofdimensiond1andd2definedbytheorthogonalunitvectorsV1a,fora=1,2,...,d1andV2b,forb=1,2,...,d2,thecosinesoftheprincipalanglesareequaltothesingularvaluesofthed1byd2matrixformedfromallthepossibledotproductsofthesetwovectors.Theresultingprincipalanglesvarybetween0andπ/2withzeroanglesappearingwhenpartsofthetwosubspacesoverlapandπ/2correspondingtodirectionsinwhichthetwosubspacesarecompletelynon‐overlapping.Theanglebetweentwosubspacesisthelargestoftheirprincipalangles.ThisdefinitionisillustratedinFigure6awhereweshowtheirregulartrajectoryofthechaoticspontaneousactivity,describedbyitstwoleadingprincipalcomponents(blackcurveinFigure6a).Thecircularorbitoftheperiodicactivity(redcurveinFigure6a)hasbeenrotatedbythesmallerofitstwoprincipalangles.Theanglebetweenthesetwosubspaces(theangledepictedinFigure6a)isthentheremaininganglethroughwhichtheperiodicorbitwouldhavetoberotatedtobringitintoalignmentwiththehorizontalplanecontainingthetwo‐dimensionalprojectionofthechaotictrajectory.

Figure6ashowstheanglebetweenthesubspacesdefinedbythefirsttwoprincipalcomponentsoftheorbitofperiodicdrivenactivityandthefirsttwoprincipalcomponentsofthechaoticspontaneousactivity.Wenowextendthisideatoacomparisonofthetwo‐dimensionalsubspaceoftheperiodicorbitandsubspacesdefinedbythefirstmprincipalcomponentsofthechaoticspontaneousactivity.ThisallowsustoseehowtheorbitliesinthefullN‐dimensionalspaceof

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neuronalactivitiesrelativetothetrajectoryofthechaoticspontaneousactivity.Theresults(Figure6b,reddots)showthatthisangleisclosetoπ/2forsmallm,equivalenttotheanglebetweentworandomlychosensubspaces.However,thevaluedropsquicklyforsubspacesdefinedbyprogressivelymoreoftheleadingprincipalcomponentsofthechaoticactivity.Ultimately,thisangleapproacheszerowhenallNofthechaoticprincipalcomponentvectorsareconsidered,asitmust,becausethesespantheentirespaceofnetworkactivities.

Figure6:Spatialpatternofnetworkresponses.a)Definitionoftheanglebetweenthesubspacedefinedbythefirsttwocomponentsofthechaoticactivity(blackcurve)andatwo‐dimensionaldescriptionoftheperiodicorbit(redcurve).b)Relationshipbetweentheorientationofperiodicandchaotictrajectories.Anglesbetweenthesubspacedefinedbythetwoprincipalcomponentsofthenon‐chaoticdrivenstateandsubspacesformedbyprincipalcomponents1throughmofthechaoticspontaneousactivity,wheremappearsonthehorizontalaxis(reddots).Blackdotsshowtheanalogousanglesbutwiththetwo‐dimensionalsubspacedefinedbyrandominputphasesreplacingthesubspaceofthenon‐chaoticdrivenactivity.c)Effectofinputfrequencyontheorientationoftheperiodicorbit.Theangle(verticalaxis)betweenthesubspacesdefinedbythetwoleadingprincipalcomponentsofnon‐chaoticdrivenactivityatdifferentfrequencies(horizontalaxis)andthesetwovectorsfora5Hzinputfrequency.d)Networkselectivitytodifferentspatialpatternsofinput.Signal(dashedcurvesandopencircles)andnoise(solidcurvesandfilledcircles)amplitudesinresponsetoinputsalignedtotheleadingprincipalcomponentsofthespontaneousactivityofthenetwork.Theinsetshowsalargerrangeonacoarserscale.Parameters:I/I1/2=0.7andf=5Hzforb,I/I1/2=1.0forc,andI/I1/2=0.2andf=2Hzford.

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Intheperiodicstate,thetemporalphasesofthedifferentneuronsdeterminetheorientationoftheorbitinthespaceofneuronalactivities.Therapidlyfallinganglebetweenthisorbitandthesubspacesdefinedbyspatialpatternsdominatingthechaoticstate(Figure6b,reddots)indicatesthatthesephasesarestronglyinfluencedbytherecurrentconnectivitythatinturndeterminesthespatialpatternofthespontaneousactivity.Asanindicationofthemagnitudeofthiseffect,wenotethattheanglesbetweentherandomphasesinusoidaltrajectoryoftheinputtothenetworkandthesamechaoticsubspacesaremuchlargerthanthoseassociatedwiththeperiodicnetworkactivity(Figure6b,blackdots).

TemporalFrequencyModulationofSpatialPatternsAlthoughrecurrentfeedbackinthenetworkplaysanimportantroleinthespatialstructureofdrivennetworkresponses,thespatialpatternoftheactivityisnotfixedbutinsteadisshapedbyacomplexinteractionbetweenthedrivinginputandtheintrinsicnetworkdynamics.Itisthereforesensitivetoboththeamplitudeandthefrequencyofthisdrive.Toseethis,weexaminehowtheorientationoftheapproximatelytwo‐dimensionalperiodicorbitofdrivennetworkactivityinthenon‐chaoticregimedependsoninputfrequency.Weusethetechniqueofprincipalanglesdescribedintheprevioussection,toexaminehowtheorientationoftheoscillatoryorbitchangeswhentheinputfrequencyisvaried.Forcomparisonpurposes,wechoosethedominanttwo‐dimensionalsubspaceofthenetworkoscillatoryresponsestoadrivinginputat5Hzasareference.Wethencalculatetheprincipalanglesbetweenthissubspaceandthecorrespondingsubspacesevokedbyinputswithdifferentfrequencies.TheresultshowninFigure6cindicatesthattheorientationoftheorbitforthesedrivenstatesrotatesastheinputfrequencychanges.

ThefrequencydependenceoftheorientationoftheevokedresponseislikelyrelatedtotheeffectseeninFigure6cinwhichhigherfrequencyactivityisprojectedontohigherprincipalcomponentsofthespontaneousactivity.Thiscausestheorbitofdrivenactivitytorotateinthedirectionofhigher‐orderprincipalcomponentsofthespontaneousactivityasthestimulusfrequencyincreases.Inaddition,thelargerthestimulusamplitude,theclosertheresponsephasesoftheneuronswillbetotherandomphasesoftheirexternalinputs(resultsnotshown).

NetworkSelectivityWehaveshownthattheresponseofanetworktorandom‐phaseinputisstronglyaffectedbythespatialstructureofspontaneousactivity(Figure6b).Wenowaskifthespatialpatternsthatdominatethespontaneousactivityinanetworkcorrespondtothespatialinputpatternstowhichthenetworkrespondsmostvigorously.Ratherthanusingrandom‐phaseinputs,wenowalignedtheinputstoournetworkalongthedirectionsdefinedbydifferentprincipalcomponentsofitsspontaneousactivity.Specifically,theinputtoneuroniissettoIViacos(2πft),whereIistheamplitudefactorandViaistheithcomponentofprincipalcomponentvectoraofthe

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spontaneousactivity.Theindexaisorderedsothata=1correspondstotheprincipalcomponentwithlargestvarianceanda=Ntheleast.Toanalyzetheresultsofusingthisinput,wedividetheresponseintoasignalcomponentcorrespondingtothetrial‐averagedresponse,andanoisecomponentconsistingofthefluctuationsaroundthisaverageresponse.Wecalltheamplitudeofthesignalcomponentoftheresponsethe“signalamplitude”andthestandarddeviationofthefluctuationsthe“noiseamplitude”.

AsseeninFigure6dtheamplitudeofthesignalcomponentoftheresponsedecreasesslowlyasafunctionofwhichprincipalcomponentisusedtodefinetheinput.Amoredramaticeffectisseenonthenoisecomponentoftheresponse.FortheinputamplitudeusedinFigure6d,inputsalignedtothefirst5principalcomponentsofthespontaneousactivitycompletelysuppressthechaoticnoise,resultinginperiodicdrivenactivity.Forhigher‐orderprincipalcomponents,thenetworkactivityischaotic.Thus,the“noise”showsmoresensitivitytothespatialstructureoftheinputthanthesignal.

DiscussionOurresultssuggestthatexperimentsthatstudythestimulus‐dependenceofthetypicallyignorednoisecomponentofresponsesshouldbeinterestingandcouldprovideinsightintothenatureandoriginofactivityfluctuations.Responsevariabilityandongoingactivityissometimesmodeledasarisingfromastochasticprocessexternaltothenetworkgeneratingtheresponses.Thisstochasticnoiseisthenaddedlinearlytothesignaltocreatethetotalneuronalactivityintheevokedstate.Ourresultsindicatethatrecurrentdynamicsofthecorticalcircuitislikelytocontributesignificantlytotheemergenceofirregularneuronalactivity,andthattheinteractionbetweensuchdeterministic“noise”andexternaldriveishighlynonlinear.Inourwork(Rajan,AbbottandSompolinsky,2009),wehaveshownthatthestimuluscausesastrongsuppressionofactivityfluctuationsandfurthermorethatthenonlinearinteractionbetweentherelativelyslowchaoticfluctuationsandthestimulusresultsinanon‐monotonicfrequencydependenceofthenoisesuppression.

Animportantfeatureofthenetworkswestudyisthatthevarianceofthesynapticstrengthsacrossthenetworkcontrolstheemergenceofinterestingcomplexdynamics.Thishasimportantimplicationsforexperimentsbecauseitsuggeststhatthemostinterestingandrelevantmodulatorsofnetworksmaybesubstancesoractivity‐dependentmodulationsthatdonotnecessarilychangepropertiesofsynapsesonaverage,butratherchangesynapticvariance.Synapticvariancecanbechangedeitherbymodifyingtherangeoverwhichsynapticstrengthsvaryacrossapopulationofsynapses,aswehavedonehere,orbymodifyingthereleaseprobabilityandvariabilityofquantalsizeatsinglesynapses.Suchmodulatorsmightbeviewedaslesssignificantbecausetheydonotchangethenetbalancebetweenexcitationandinhibition.However,networkmodelingsuggeststhatsuchmodulationsareofgreatimportanceincontrollingthestateoftheneuronalcircuit.

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Therandomcharacteroftheconnectivityinournetworkprecludesasimpledescriptionofthespatialactivitypatternsintermsoftopographicallyorganizedmaps.Ouranalysisshowsthatevenincorticalareaswheretheunderlyingconnectivitydoesnotexhibitsystematictopography,dissectingthespatialpatternsoffluctuationsinneuronalactivitycanrevealimportantinsightaboutbothintrinsicnetworkdynamicsandstimulusselectivity.Principalcomponentanalysisrevealedthatdespitethefactthatthenetworkconnectivitymatrixisfullrank,theeffectivedimensionalityofthechaoticfluctuationsismuchsmallerthanthenumberofneuronsinthenetwork.Thissuppressionofspatialmodesismuchstrongerthanexpectedfromalinearnetworklow‐passfilteringaspatio‐temporalwhitenoiseinput.Furthermore,asinthetemporaldomain,activespatialpatternsexhibitstrongnonlinearinteractionbetweenexternaldrivinginputsandintrinsicdynamics.Surprisingly,evenwhenthestimulusamplitudeisstrongenoughtofullyentrainthetemporalpatternofnetworkactivity,spatialorganizationoftheactivityisstillstronglyinfluencedbyrecurrentdynamics,asshowninFigures6cand6d.

Wehavepresentedtoolsforanalyzingthespatialstructureofchaoticandnon‐chaoticpopulationresponsesbasedonprincipalcomponentanalysisandanglesbetweentheresultingsubspaces.Principalcomponentanalysishas,beenappliedprofitablytoneuronalrecordings(see,forexample,Broome,JayaramanandLaurent,2006).Theseanalysesoftenplotactivitytrajectoriescorrespondingtodifferentnetworkstatesusingthefixedprincipalcomponentcoordinatesderivedfromcombinedactivitiesunderallconditions.Ouranalysisoffersacomplementaryapproachwherebyprincipalcomponentsarederivedforeachstimulusconditionseparately,andprincipalanglesareusedtorevealnotonlythedifferencebetweentheshapesoftrajectoriescorrespondingtodifferentnetworkstates,butalsothedifferenceintheorientationofthelowdimensionalsubspacesofthesetrajectorieswithinthefullspaceofneuronalactivity.

Manymodelsofselectivityincorticalcircuitsrelyonknowledgeofthespatialorganizationofafferentinputsaswellascorticalconnectivity.However,inmanycorticalareas,suchinformationisnotavailable.Ourresultsshowthatexperimentallyaccessiblespatialpatternsofspontaneousactivity(e.g.fromvoltage‐orcalcium‐sensitiveopticalimagingexperiments)canbeusedtoinferthestimulusselectivityinducedbythenetworkdynamicsandtodesignspatiallyextendedstimulithatevokestrongresponses.Thisisparticularlytruewhenselectivityismeasuredintermsoftheabilityofastimulustoentraintheneuraldynamics,asinFigure6d.Ingeneral,ourresultsindicatethattheanalysisofspontaneousactivitycanprovidevaluableinformationaboutthecomputationalimplicationsofneuronalcircuitry.

AcknowledgmentsResearchofKRandLAsupportedbyNationalScienceFoundationgrantIBN‐0235463andanNIHDirector'sPioneerAward,partoftheNIHRoadmapforMedicalResearch,throughgrantnumber5‐DP1‐OD114‐02.HSispartiallysupportedbygrantsfromtheIsraelScienceFoundationandtheMcDonnell

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Foundation.ThisresearchwasalsosupportedbytheSwartzFoundationthroughtheSwartzCentersatColumbiaandHarvardUniversities.

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